Element Displacement Functions Research Articles

A Double Thick Beam FEM with Explosion Resistance Dynamics Analysis Based on Spline Wavelets in Underground Structure

Isoparametric B-spline wavelet functions can provide excellent approximation for smooth functions. In this paper, the response of composite laminated plate subjected to explosions is analyzed by using a double thick beam FEM mode based on the spline wavelet function. This mode, which can be considered in the distribution of normal stress and transverse shear stress simultaneously, is very important in the analysis of thick plate or blast-damage mechanism. The mode in analyzing underground structure and earth medium respond has the advantage of high precision and fast constringency and is easy to establish a element displacement function. The comparison between simulative and experimental results shows the effectiveness and precision of this method.

Improved Finite-Segment Method for Analyzing Shear Lag Effect in Thin-Walled Box Girders

AbstractShear lag effect in thin-walled box girders has been studied over several decades. However, the methods adopted in many papers have some deficiencies. In the present work, an improved displacement function for shear lag warping in a box girder with cantilever slabs is established. Based on the concept of generalized force corresponding to the generalized displacement for shear lag and the relevant geometrical properties, an improved finite-segment method is proposed to simplify the shear lag analysis of complex box girders. The homogeneous solution of the governing differential equation for shear lag is adopted as the element displacement function. The formulas of the element stiffness matrix and the equivalent nodal force vector are derived. A general formula expressed in terms of the generalized moment is presented to calculate the stress. A finite-element computer program is developed by using FORTRAN language and is used to analyze a cantilever box girder model and a continuous prestressed con...

A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

AbstractC0−1 patch test (Int. J. Numer. Meth. Engng 2004; 61:433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C0−1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C0−1 patch test, the element displacement functions used to calculate strains must satisfy C0 continuity (or weak C0 continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C1 continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C0 continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C0−1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.

FEM-based approach for the stability design of thin-walled members by using cFSM base functions

This paper presents a new design approach for the stability design of thin-walled members. The proposed approach is based on the buckling modes and critical forces/moments determined by a linear buckling analysis performed on a regular shell finite element model. A fully automatic buckling mode identification technique is applied, by using the modal base functions of the newly proposed constrained finite strip method, where the various buckling types are separated by clearly defined mechanical criteria. The paper briefly summarizes the determination of modal base functions which then are used to approximate finite element displacement functions (i.e., buckling modes). The mode identification method provides the lowest critical values (forces or moments) to all the three characteristic buckling types: global, distortional and local, on the basis of which the buckling resistance can be calculated by using the design formulae of the direct strength method. The proposed new approach, which is potentially more general than any of the existing design approaches, is demonstrated on Z columns and beams with simple loading and boundary conditions. Critical values as well as resistances are calculated for some selected cases, the results are compared to those of another design method. The comparisons prove the applicability of the proposed procedure. Further research is necessary to extend the proposal for more general and more complex practical cases.

Refined 15‐DOF triangular discrete degenerated shell element with high performances

AbstractA refined discrete degenerated 15‐DOF triangular shell element RDTS15 with high performances is proposed. For constructing the element displacement function, the exact displacement function of the Timoshenko's beam is used as the displacement on the element boundary, and the re‐constitute method for shear strain matrix is adopted. The proposed element can be used in the analysis of both moderate thick and thin plates/shells. Numerical examples presented show that the new model indeed possesses higher accuracy in the analysis of thin and thick plates/shells, and that it can pass the patch test required for the Kirchhoff thin plate elements, and also passed the inf–sup test for free cylindrical shell problems and satisfied both the bending‐ and membrane‐dominated test. Copyright © 2004 John Wiley Sons, Ltd.

A refined discrete triangular Mindlin element for laminated composite plates

Based on the Mindlin plate theory, a refined discrete 15-DOF triangular laminated composite plate finite element RDTMLC with the re-constitution of the shear strain is proposed. For constituting the element displacement function, the exact displacement function of the Timoshenko's laminated composite beam as the displacement on the element boundary is used to derive the element displacements. The proposed element can be used for the analysis of both moderately thick and thin laminated composite plate, and the convergence for the very thin situation can be ensured theoretically. Numerical examples presented show that the present model indeed possesses the properties of higher accuracy for anisotropic laminated composite plates and is free of locking even for extremely thin laminated plates.

Boundary stress calculation—a comparison study

This paper investigates the accuracy of boundary stress evaluation by the boundary element method. Three approaches are presented which are all based on post-processing to recover the boundary stresses. The first approach is the standard method which employs the derivative of the element displacement function and Hooke’s law to calculate boundary stresses. The second approach is the hypersingular integral method which derives the stresses from the derivative of the original boundary element formulation. The last approach uses the stress continuity conditions to obtain a high order stress function which is continuous over the interelement node. The fundamental formulations for the three methods are reviewed in the paper, and four numerical examples are presented to compare their accuracy on boundary tangential stress evaluation. The influence of mesh size on the stress accuracy is also studied.

AN IMPROVED PROCEDURE FOR THE DETERMINATION OF HOT SPOT STRESSES IN TUBULAR JOINTS

Abstract— A conventional finite element method may show a weakness when determining the hot spot stress distributions in the brace/chord intersection region of offshore tubular joints. This is because the chosen element displacement functions do not implicitly satisfy the conditions which prevail on the free surfaces. A procedure has been proposed to modify the conventional finite element method so as to allow the hot spot stresses, which occur at the free boundary of the weld toe of tubular joints, to be determined with improved accuracy. The results obtained by this modified method are compared with both an experimental and a traditional finite element solution. The comparison shows that the modified solution is in better agreement with the experimental data as compared with the traditional solution.

Natural frequencies of plates with square holes when subjected to in-plane uniaxial, biaxial or shear loading

The finite element method is used to determine the natural frequencies of flat square plates containing eccentrically located square holes. The plates are subjected to in-plane uniaxial, biaxial or uniformly distributed shear along the four outer edges. These edges are either simply supported or clamped. In order to evaluate the stiffness and mass matrices, the non-conforming rectangular displacement element is used to model the out-of-plane behaviour of the plate. The in-plane stress distribution within the plate, required in the analysis, is determined by using a rectangular finite element having the only essential degrees of freedom at each of the four corner nodes. The element displacement functions are based on assumed strains and satisfy the exact requirements of strain free rigid-body modes of displacements. The variation of the natural frequency with the size and location of the hole is first investigated in the absence of any in-plane stresses. This analysis is then repeated for a series of values of the applied in-plane stresses. When uniform shear is applied, tension and compression zones are produced in the plate and hence the effect of locating the hole in each of the regions is also investigated. The values of the applied in-plane stresses ranged up to the point that would cause the plates to buckle. In this way a comprehensive set of results can be obtained.

Geometric nonlinear analysis by using refined triangular thin plate element and free form membrane locking

Based on a large deformation variational principle with relaxed interelement continuity requirement in the total Lagranginan description, a refined triangular thin plate element for geometric nonlinear analysis has been developed. By introducing special element displacement functions into the geometric stiffness matrix, the membrane locking phenomenon is relieved effectively. The numerical results are presented to show that the present element possesses higher accuracy and an ability to free form membrane locking.

Determination of Inter‐ and Free‐Boundary Stresses in Plate Analysis

The conventional finite‐element method shows its weakness in determining the stress distributions along free‐ and interboundaries for plate (facet shell) analysis. This is because the chosen element displacement and rotation functions do not implicitly satisfy the conditions that prevail at the boundaries. A procedure has been proposed to modify the conventional finite‐element method so as to allow the elastic stresses along the free‐ and interboundaries of plates to be determined with improved accuracy. The results obtained by this modified method are compared with both an experimental and a traditional finite‐element solution. The comparison shows that the modified solution is in better agreement with the experimental test data as compared with the traditional solution. The computation time required to perform the additional calculations for determining free‐ and interboundary stresses with improved accuracy is only a small fraction of that required for the complete finite‐element analysis.

Improved Method for Determining Interfacial Stresses in Composites

The conventional finite element method shows its weakness in determining the stress distributions along fiber matrix interfaces. This is because the chosen element displacement functions do not explicitly and/or implicitly satisfy the equilibrium and compatibility conditions that prevail at the interfaces. A procedure has been proposed to modify the conventional finite element method so as to allow the elastic stresses along the fiber matrix interfaces of two‐ and three‐dimensional fiber‐reinforced composites to be determined with improved accuracy. The results obtained by this modified method for a two‐dimensional problem are compared with both a photoelastic and a traditional finite elelment solution. The comparison shows that the modified solution is in better agreement with the photoelastic test data than the traditional solution. The results obtained by the modified method for a three‐dimensional problem are compared with a traditional finite element solution and the modified method is found to be superior.

An improved method for determining free boundary stresses

The conventional finite element method shows its weakness in determining the stress distributions along free boundaries. This is because the chosen element displacement functions do not implicitly satisfy the conditions which prevail at the free boundaries. A procedure has been proposed to modify the conventional finite element method so as to allow the elastic stresses along the free boundaries of two- and three-dimensional structures to be determined with improved accuracy. The results obtained by this modified method for a two-dimensional problem are compared with both a photoelastic and a traditional finite element solution. The comparison shows that the modified solution is in better agreement with the photoelastic test data as compared with the traditional solution. The results obtained by the modified method for a three-dimensional problem is compared with a traditional finite element solution and the former is found to be superior.

P‐version hierarchical three dimensional curved shell element for elastostatics

AbstractThis paper presents a hierarchical three dimensional curved shell finite element formulation based on the p‐approximation concept. The element displacement approximation can be of arbitrary and different polynomial orders in the plane of the shell (ξ, η) and the transverse direction (ξ). The curved shell element approximation functions and the corresponding nodal variables are derived by first constructing the approximation functions of orders pξ, pη and pξ and the corresponding nodal variable operators for each of the three directions ξ, η and ξ and then taking their products (sometimes also known as tensor product). This procedure gives the approximation functions and the corresponding nodal variables corresponding to the polynomial orders pξ, pη and pξ. Both the element displacement functions and the nodal variables are hierarchical; therefore, the resulting element matrices and the equivalent nodal load vectors are hierarchical also, i.e. the element properties corresponding to the polynomial orders pξ, pη and pξ are a subset of those corresponding to the orders (pξ + 1), (pη +1) and (pξ +1). The formulation guarantees C° continuity or smoothness of the displacement field across the interelement boundaries.The geometry of the element is described by the co‐ordinates of the nodes on its middle surface (ξ = 0) and the nodal vectors describing its bottom (ξ = −1) and top (ξ = +1) surfaces. The element properties are derived using the principle of virtual work and the hierarchical element approximation. The formulation is equally effective for very thin as well as very thick plates and curved shells. In fact, in many three dimensional applications the element can be used to replace the hierarchical three dimensional solid element without loss of accuracy but significant gain in modelling convenience. Numerical examples are presented to demonstrate the accuracy, efficiency and overall superiority of the present formulation. The results obtained from the present formulation are compared with those available in the literature as well as analytical solutions.

New triangular finite element for plates and shells

A new triangular shell finite element ‘TNTE.1’ (Ten Node Triangular Element, Model 1) is presented. The formulation is based on Sanders' theory which involves the inclusion of the normal rotation Φn in the bending‐strain relations only. The element displacement functions are complete cubic polynomials for inplane displacements u and v. For out‐of‐plane displacement w, three new singular rational shape functions were added at the element corners. Thus a conforming triangular element with twenty seven degrees‐of‐freedom is obtained after eliminating the internal displacements by static condensation. The formulation of this element is new in that an integration technique is developed and applied to the element stiffness matrix and load vector. This technique is based on performing all the necessary integrations externally (i.e. outside the main computer program) and then modifying the formulation of the element matrices to account for this change. Hence, such a method allows the inclusion of higher‐order integration rules without any loss of economy, due to computer time, in the main program. Results using this element showed good agreement with other finite element and closed form solutions.

The strain approach to the development of thin cylindrical shell finite elements

The method of forming thin cylindrical shell finite element displacement functions by initially developing the strain function is presented. The Kirchhoff-Love assumptions are used together with deep thin shell finite elements to develop a number of 20 degrees-of-freedom rectangular and 15 degrees-of-freedom triangular cylindrical shell elements. The elements are tested on an extended range of barrel vault roof problems and results are presented. Only in the very specific case of N φ for the free straight-edged barrel vault using the triangular elements are the results found to be inaccurate. In all cases where the displacements and stresses can be analytically checked, the strain approach gives accurate results.

Singular finite element efficiency in estimating stress intensity factors

AbstractSingular finite element displacement functions for linear elastic fracture mechanics applications are investigated. The displacement fields are expressed in terms of polynomials. These polynomials are expanded in varying degrees of r and θ in order to examine the effect of the number and type of terms on the solution to various problems with known solutions. Methods of estimating the stress intensity factors for mode 1 and mode 2 are presented. Numerical results evaluating the effect of various r and θ terms on the solutions are presented.

A combined Rayleigh-Ritz/finite element method for the nonlinear analysis of stiffened plated structures

The paper describes a hybrid method for the non-linear analysis of steel plates and stiffened plating in which local (finite element) displacement functions are supplemented or replaced by global functions. The method is applied to the collapse analysis of box-girder bridges.

Explicit addition of rigid-body motions in curved finite elements.

Several aspects of the representation of rigid-body modes in finite element displacement functions are discussed. First, a simple technique to compute six independent modes is demonstrated; it offers the possibility to estimate the extent to which the rigid modes are included in the specified interpolation functions; it also enables one to check the strain displacement relationships. Then, a procedure to find out how many and which rigid modes are present in a stiffness matrix is explained. Finally, the missing modes are explicitly added and the associated degrees of freedom are condensed out; in most cases the operation has the effect of rendering the elements incompatible. A doubly curved quadrilateral element for shells of revolution with 24° of freedom is used to illustrate these effects; several examples are solved with the element, demonstrating the advantages and drawbacks of explicitly adding rigid modes to an element.

Rectangular finite elements for analysis of plate vibrations

The vibration of flat rectangular plates has been investigated using rectangular finite elements. Three different element displacement functions were used to determine the effects of increasing the degree of continuity between adjacent elements. The first function gave continuity of displacement and slope, the second also had continuity of twist and the third had continuity of displacement, slope, twist and curvature. These functions were used to calculate the natural frequencies and modes of rectangular plates with clamped and simply-supported edge constraints. A comparison of the results showed that the second function gave the most satisfactory results. The calculated natural frequencies were found to be much more accurate than those obtained using element displacement functions giving only displacement continuity between adjacent elements.